Kodaira's table of singular fibers has a singular fiber for each of $\tilde{A}_n$, $\tilde{D}_n$, $\tilde{E}_6$, $\tilde{E}_7$, and $\tilde{E}_8$; these are chains or cycles of (-2)-curves connected to each other based on the aforementioned Dynkin diagram.

On the other hand $A_n$, $D_n$, $E_6$, $E_7$, and $E_8$ Dynkin diagrams, by McKay correspondence, describe the exceptional curves of quotient singularities; i.e. if $G$ is a finite group corresponding to one of these Dynkin diagrams, $\mathbb{C}^2/G$ has a resolution whose exceptional curve is equal to that Dynkin diagram.

Question: Is there a similar construction of Kodaira's singular fibers? i.e. can we obtain them as a resolution of some quotient prescribed by one of $\tilde{A}_n$, $\tilde{D}_n$, $\tilde{E}_6$, $\tilde{E}_7$, and $\tilde{E}_8$.


1 Answer 1


If you take the Tate curve $\mathbb G_m/\langle q \rangle$ over $\mathbb Z[[q]]$ and mod out by the subgroup $\mu_n$ and resolve singularities, you should get a copy of the semistable fiber type $I_n$ with lattice $A_{n-1}$. Note that this means the Tate curve, modulo nothing, is of fiber type $I_1$, and hence already has a singular fiber.

However $I_n$ can also be expressed in a number of other ways, like as a cover of $I_1$.

The fibers $I_n^*$ can be expressed as the quotient of $I_{2n}$ by the automorphism sending $q \to -q$ and inverting $\mathbb G_m$. So combining this with the previous we can write it as a quotient of $I_1$ by a $D_{2n}$ subgroup, or $I_2$ by a $D_n$.

The remaining fibers can be viewed as resolutions of a quotient of a family of smooth elliptic curves. One takes a constant elliptic curve $E$ over a curve with coordinate $t$ and an automorphism $\sigma$ of $E$ of order and takes the quotient by the map $E \to \sigma(E)$, $t \to \mu t$ for $\mu$ a fixed $n$th root of unity. The different automorphisms and roots of unity will give all the remaining fiber types.

So one can express the different fibers as quotients, but as quotients of different objects, and the groups do not match up as well as in the McKay correspondence.

Perhaps there is a better way of doing it that avoids these problems, but I don't see a particular reason for one to exist.


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